Water flows vertically upward in a circular cross-sectional...

Water flows vertically upward in a circular cross-sectional pipe. At section $(1),$ the velocity profile over the cross-sectional area is uniform. At section (2), the velocity profile is \[ \mathbf{V}=w_{c}\left(\frac{R-r}{R}\right)^{1 / 7} \hat{\mathbf{k}} \] where $\mathbf{V}=$ local velocity vector, $w_{c}=$ centerline velocity in the axial direction, $R=$ pipe inside radius, and $r=$ radius from pipe axis. Develop an expression for the loss in available energy between sections (1) and (2)

Water flows vertically upward in a circular cross-sectional pipe. At section $(1),$ the velocity profile over the cross-sectional area is uniform. At section (2), the velocity profile is
\[
\mathbf{V}=w_{c}\left(\frac{R-r}{R}\right)^{1 / 7} \hat{\mathbf{k}}
\]
where $\mathbf{V}=$ local velocity vector, $w_{c}=$ centerline velocity in the axial direction, $R=$ pipe inside radius, and $r=$ radius from pipe axis. Develop an expression for the loss in available energy between sections (1) and (2)

Key Concepts

Effects of Non-uniform Flow on Available Energy

Non-uniform flow introduces spatial variations in kinetic energy, which can lead to a loss in available energy between different sections of a pipe. These losses occur because a highly concentrated kinetic energy region (for instance, near the center of the pipe) is less efficiently converted to pressure energy than in a uniform profile. Quantifying this loss requires comparing the energy computed using a uniform velocity assumption with that obtained by accounting for the actual distribution of velocities.

Velocity Profile Integration over a Cross?Section

When a velocity profile varies across a pipe’s cross-section, integral calculus is used to derive mean velocities and kinetic energy distributions. By integrating the local velocity or its powers over the area, one obtains the average flow characteristics, which are then used to compute key quantities such as flow rate and kinetic energy correction factors. These computations are essential for establishing accurate energy balances in fluid systems.

Bernoulli’s Principle and Energy Conservation

This concept involves the conservation of mechanical energy in fluid flow. It states that the sum of the pressure energy, potential energy, and kinetic energy remains constant along a streamline in an inviscid, steady flow. When analyzing flows in pipes, Bernoulli’s equation is modified to account for losses and non?uniform velocity distributions, which can lead to differences in available energy between sections of the flow.

Kinetic Energy Correction Factor

In flows with non?uniform velocity profiles the actual distribution of kinetic energy is not adequately captured by simply using the square of the average velocity. The kinetic energy correction factor, often denoted by ?, quantifies the discrepancy between the mean kinetic energy computed from the local velocity profile and the energy computed using the average velocity. This factor is crucial when determining the available energy head in systems where the velocity distribution deviates from uniformity.

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